Subgroups — #27173961

Order 12928.

928 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108075	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15692217	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265084	C₂	1	1	yes	maximal, characteristic, complemented
5	#20265055	C₂	1	2	yes	maximal, complemented
6	#20265055	C₂	1	2	yes	maximal, complemented
7	#20265039	C₂	1	2	yes	maximal, complemented
8	#20265034	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13108051	C₂	1	1	yes	maximal, characteristic, complemented
10	#95×D₁₀₁	C₂	1	2	yes	maximal, complemented
11	#391×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13108075	—	2	4	no	—
13	C₂×#13108075	—	2	4	no	—
14	C₂×#13108075	C₄	1	8	yes	complemented
15	C₂²×D₄₀₄	C₂²	1	2	yes	—
16	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
18	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
19	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
20	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
21	C₄×D₄₀₄	—	2	4	no	—
22	#15692217	—	2	4	no	—
23	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×#13108051	C₂²	1	1	yes	characteristic
25	C₂×#13108051	C₂²	1	1	yes	characteristic
26	C₂×#13108051	C₂²	1	2	yes	—
27	C₂×#13108051	C₂²	1	2	yes	—
28	C₂×#13108051	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
30	#15692162	C₂²	1	1	yes	characteristic
31	#15692162	C₂²	1	2	yes	—
32	#15692162	C₂²	1	2	yes	—
33	#15692162	C₂²	1	1	yes	characteristic
34	#15692162	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
38	#95×C₁₀₁	C₂²	1	2	yes	—
39	#95×C₁₀₁	C₂²	1	2	yes	—
40	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
41	C₂³×D₁₀₁	C₂³	1	2	yes	—
42	#13108075	—	2	32	no	—
43	C₂×D₄₀₄	—	2	4	no	—
44	C₂×D₄₀₄	—	2	4	no	—
45	C₂×D₄₀₄	C₂×C₄	1	4	yes	—
46	C₂×D₄₀₄	C₂×C₄	1	8	yes	complemented
47	C₂×D₄₀₄	D₄	1	4	yes	—
48	C₂×D₄₀₄	D₄	1	4	yes	—
49	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
50	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
51	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
52	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₁₀₁	—	2	4	no	—
54	C₂×C₄×D₁₀₁	—	2	4	no	—
55	C₂×C₄×D₁₀₁	—	2	2	no	—
56	C₂×C₄×D₁₀₁	—	2	2	no	—
57	C₂×C₄×D₁₀₁	—	2	4	no	—
58	C₂×C₄×D₁₀₁	—	2	4	no	—
59	C₂×C₄×D₁₀₁	—	2	4	no	—
60	C₂×C₄×D₁₀₁	—	2	4	no	—
61	C₂×C₄×D₁₀₁	—	4	2	no	—
62	C₂×C₄×D₁₀₁	—	4	2	no	—
63	C₂×C₄×D₁₀₁	C₂×C₄	1	8	yes	complemented
64	C₂×C₄×D₁₀₁	C₂×C₄	1	4	yes	complemented
65	C₄⋊Dic₁₀₁	—	2	2	no	—
66	#13108051	—	2	4	no	—
67	#13108051	—	2	4	no	—
68	#13108051	—	2	2	no	—
69	#13108051	—	2	2	no	—
70	#13108051	—	4	2	no	—
71	C₄×Dic₁₀₁	—	2	2	no	—
72	C₂×Q₈×C₁₀₁	—	2	2	no	—
73	C₂×Q₈×C₁₀₁	—	2	2	no	—
74	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	complemented
75	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
76	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
78	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₀₁	—	2	2	no	—
81	C₄²×C₁₀₁	—	2	2	no	—
82	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
83	C₂²×D₁₀₁	—	2	2	no	—
84	C₂²×D₁₀₁	—	2	2	no	—
85	C₂²×D₁₀₁	—	2	2	no	—
86	C₂²×D₁₀₁	—	2	2	no	—
87	C₂²×D₁₀₁	—	2	1	no	—
88	C₂²×D₁₀₁	—	2	1	no	—
89	C₂²×D₁₀₁	—	2	1	no	—
90	C₂²×D₁₀₁	—	2	1	no	—
91	C₂²×D₁₀₁	—	2	1	no	—
92	C₂²×D₁₀₁	—	2	1	no	—
93	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
94	C₂²×D₁₀₁	C₂²×C₄	1	4	yes	—
95	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
96	C₂²×D₁₀₁	C₂×D₄	1	4	yes	—
97	Q₈×C₁₀₁	—	2	8	no	—
98	C₂×C₄×C₁₀₁	—	2	1	no	—
99	C₂×C₄×C₁₀₁	—	2	2	no	—
100	C₂×C₄×C₁₀₁	—	2	2	no	—
101	C₂×C₄×C₁₀₁	—	2	1	no	—
102	C₂×C₄×C₁₀₁	—	2	1	no	—
103	C₂×C₄×C₁₀₁	—	2	2	no	—
104	C₂×C₄×C₁₀₁	—	2	1	no	—
105	C₂×C₄×C₁₀₁	—	2	2	no	—
106	C₂×C₄×C₁₀₁	—	2	2	no	—
107	C₂×C₄×C₁₀₁	—	2	1	no	—
108	C₂×C₄×C₁₀₁	C₂²×C₄	1	4	yes	—
109	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₀₁	#41	1	2	yes	complemented
113	C₂×Dic₁₀₁	—	2	1	no	—
114	C₂×Dic₁₀₁	—	2	1	no	—
115	C₂×Dic₁₀₁	—	2	1	no	—
116	C₂×Dic₁₀₁	—	2	1	no	—
117	C₂×Dic₁₀₁	—	2	1	no	—
118	C₂×Dic₁₀₁	—	2	2	no	—
119	C₂×Dic₁₀₁	—	2	1	no	—
120	C₂×Dic₁₀₁	—	2	2	no	—
121	C₂×Dic₁₀₁	—	2	2	no	—
122	C₂×Dic₁₀₁	—	2	1	no	—
123	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
124	C₂×Dic₁₀₁	#41	1	2	yes	complemented
125	D₄₀₄	—	2	16	no	—
126	D₄₀₄	C₂²⋊C₄	1	16	yes	complemented
127	C₄×D₁₀₁	—	2	16	no	—
128	C₄×D₁₀₁	—	2	8	no	—
129	C₄×D₁₀₁	—	4	8	no	—
130	C₄×D₁₀₁	—	4	8	no	—
131	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
132	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
133	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
134	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
135	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
136	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂²×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
138	C₂×D₁₀₁	—	2	4	no	—
139	C₂×D₁₀₁	—	2	4	no	—
140	C₂×D₁₀₁	—	2	4	no	—
141	C₂×D₁₀₁	—	2	4	no	—
142	C₂×D₁₀₁	—	2	4	no	—
143	C₂×D₁₀₁	—	2	4	no	—
144	C₂×D₁₀₁	—	2	2	no	—
145	C₂×D₁₀₁	—	2	2	no	—
146	C₂×D₁₀₁	—	2	2	no	—
147	C₂×D₁₀₁	—	4	1	no	—
148	C₂×D₁₀₁	—	4	1	no	—
149	C₂×D₁₀₁	—	4	1	no	—
150	C₂×D₁₀₁	—	4	1	no	—
151	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	8	yes	—
152	C₄×C₁₀₁	—	2	4	no	—
153	C₄×C₁₀₁	—	2	2	no	—
154	C₄×C₁₀₁	—	4	2	no	—
155	C₄×C₁₀₁	—	4	1	no	—
156	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
157	Dic₁₀₁	—	2	2	no	—
158	Dic₁₀₁	—	2	2	no	—
159	Dic₁₀₁	—	4	2	no	—
160	Dic₁₀₁	—	4	1	no	—
161	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₁₀₁	C₄×#41	1	1	yes	characteristic
163	C₂×C₁₀₁	#520	1	1	yes	characteristic
164	C₂×C₁₀₁	#546	1	1	yes	characteristic
165	C₂×C₁₀₁	#546	1	1	yes	characteristic
166	C₂×C₁₀₁	#547	1	1	yes	characteristic
167	C₂×C₁₀₁	#547	1	1	yes	characteristic
168	D₁₀₁	—	2	8	no	—
169	D₁₀₁	—	4	2	no	—
170	#2339	—	101	1	no	maximal, Sylow
171	C₁₀₁	#2339	1	1	yes	characteristic, complemented, Sylow
172	C₂²×#41	—	101	1	no	—
173	C₂×C₄×D₄	—	101	1	no	—
174	C₂×C₄×D₄	—	101	1	no	—
175	#391	—	101	1	no	—
176	#391	D₁₀₁	1	1	yes	characteristic, complemented
177	#386	—	101	2	no	—
178	#386	—	101	2	no	—
179	#381	—	101	2	no	—
180	#380	—	101	1	no	—
181	C₄×C₂²⋊C₄	—	101	1	no	—
182	C₂×#95	—	101	2	no	—
183	C₂×#41	—	101	8	no	—
184	C₂×#41	—	202	4	no	—
185	C₂×#41	—	202	4	no	—
186	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
187	C₂²×D₄	—	101	1	no	—
188	C₂²×D₄	—	101	2	no	—
189	C₂³×C₄	—	101	2	no	—
190	C₂³×C₄	—	101	1	no	—
191	C₂³×C₄	—	101	2	no	—
192	C₂³×C₄	—	101	2	no	—
193	C₄×D₄	—	202	4	no	—
194	C₄×D₄	—	202	4	no	—
195	C₂×C₄⋊C₄	—	101	1	no	—
196	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
197	C₂×C₂²⋊C₄	—	101	2	no	—
198	C₂×C₂²⋊C₄	—	101	2	no	—
199	C₂×C₂²⋊C₄	—	101	2	no	—
200	C₂×C₂²⋊C₄	—	101	1	no	—
201	C₂×C₂²⋊C₄	—	101	1	no	—
202	C₂×C₄²	—	101	1	no	—
203	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
204	#95	—	101	1	no	—
205	#95	—	101	2	no	—
206	#95	—	101	2	no	—
207	#95	—	101	1	no	—
208	#95	—	101	2	no	—
209	#95	C₂×D₁₀₁	1	2	yes	—
210	#95	C₂×D₁₀₁	1	2	yes	—
211	C₂⁴	—	101	2	no	—
212	C₂⁴	—	101	1	no	—
213	#41	—	202	32	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	C₄×D₁₀₁	1	4	yes	complemented
217	C₂×D₄	—	101	4	no	—
218	C₂×D₄	—	101	8	no	—
219	C₂×D₄	—	101	4	no	—
220	C₂×D₄	—	101	4	no	—
221	C₂×D₄	—	202	4	no	—
222	C₂×D₄	—	202	4	no	—
223	C₂²×C₄	—	101	1	no	—
224	C₂²×C₄	—	101	1	no	—
225	C₂²×C₄	—	101	8	no	—
226	C₂²×C₄	—	101	4	no	—
227	C₂²×C₄	—	101	1	no	—
228	C₂²×C₄	—	101	2	no	—
229	C₂²×C₄	—	404	2	no	—
230	C₂²×C₄	—	404	2	no	—
231	C₂²×C₄	—	202	4	no	—
232	C₂²×C₄	—	202	4	no	—
233	C₂²×C₄	—	202	4	no	—
234	C₂²×C₄	—	202	4	no	—
235	C₂²×C₄	—	202	4	no	—
236	C₂²×C₄	—	202	4	no	—
237	C₂²×C₄	—	202	2	no	—
238	C₂²×C₄	—	202	2	no	—
239	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
240	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
241	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
242	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	202	2	no	—
246	C₂²⋊C₄	—	404	2	no	—
247	C₂²⋊C₄	—	202	2	no	—
248	C₂²⋊C₄	—	202	4	no	—
249	C₂²⋊C₄	—	202	4	no	—
250	C₂²⋊C₄	—	202	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	202	2	no	—
253	C₂³	—	101	4	no	—
254	C₂³	—	101	4	no	—
255	C₂³	—	101	4	no	—
256	C₂³	—	101	2	no	—
257	C₂³	—	202	1	no	—
258	C₂³	—	202	1	no	—
259	C₂³	—	202	1	no	—
260	C₂³	—	202	1	no	—
261	C₂³	—	202	1	no	—
262	C₂³	—	202	1	no	—
263	C₂³	—	202	2	no	—
264	C₂³	—	202	2	no	—
265	C₂³	—	202	2	no	—
266	C₂³	—	202	2	no	—
267	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
268	Q₈	—	2	8	no	—
269	D₄	—	101	16	no	—
270	D₄	—	202	16	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	101	2	no	—
282	C₂×C₄	—	101	2	no	—
283	C₂×C₄	—	404	8	no	—
284	C₂×C₄	—	404	8	no	—
285	C₂×C₄	—	202	16	no	—
286	C₂×C₄	—	202	1	no	—
287	C₂×C₄	—	202	2	no	—
288	C₂×C₄	—	202	1	no	—
289	C₂×C₄	—	202	2	no	—
290	C₂×C₄	—	202	2	no	—
291	C₂×C₄	—	202	1	no	—
292	C₂×C₄	—	202	1	no	—
293	C₂×C₄	—	202	1	no	—
294	C₂×C₄	—	202	1	no	—
295	C₂×C₄	—	202	1	no	—
296	C₂×C₄	—	202	8	no	—
297	C₂×C₄	C₂×C₄×D₁₀₁	1	4	yes	—
298	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
299	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
300	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
301	C₂×C₄	#13108075	1	2	yes	complemented
302	C₂²	—	101	8	no	—
303	C₂²	—	404	1	no	—
304	C₂²	—	404	1	no	—
305	C₂²	—	404	1	no	—
306	C₂²	—	404	1	no	—
307	C₂²	—	202	2	no	—
308	C₂²	—	202	2	no	—
309	C₂²	—	202	2	no	—
310	C₂²	—	202	4	no	—
311	C₂²	—	202	4	no	—
312	C₂²	—	202	4	no	—
313	C₂²	—	202	4	no	—
314	C₂²	—	202	4	no	—
315	C₂²	—	202	4	no	—
316	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
317	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
318	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
319	C₂²	#41×D₁₀₁	1	1	yes	characteristic
320	C₂²	C₂×#13108075	1	1	yes	characteristic
321	C₂²	#15692301	1	1	yes	characteristic
322	C₂²	#15692303	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	4	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	1	no	—
327	C₄	—	404	2	no	—
328	C₄	—	404	1	no	—
329	C₄	—	202	2	no	—
330	C₄	—	202	2	no	—
331	C₄	C₂²⋊C₄×D₁₀₁	1	4	yes	—
332	C₂	—	404	2	no	—
333	C₂	—	202	8	no	—
334	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
335	C₂	C₄×#13108075	1	1	yes	characteristic
336	C₂	#20265659	1	1	yes	characteristic
337	C₂	#20265818	1	1	yes	characteristic
338	C₂	#20265820	1	1	yes	characteristic
339	C₂	#20265829	1	1	yes	characteristic
340	C₂	#20265830	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Fitting	#391×C₁₀₁	C₂
socle, cosocle	C₂³×C₁₀₁	C₂⁴
prime radical	C₂²×#13108075	C₂
hypercenter	#391	D₁₀₁
hypocenter	C₁₀₁	#2339
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #391

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₄×D₄₀₄	C₂	semidirect
3. C₂×#15692217	C₂	semidirect
4. #20265084	C₂	semidirect
5. #20265055	C₂	semidirect
6. #20265055	C₂	semidirect
7. #20265039	C₂	semidirect
8. #20265034	C₂	semidirect
9. C₄×#13108051	C₂	semidirect
10. #95×D₁₀₁	C₂	semidirect
11. #391×C₁₀₁	C₂	semidirect
14. C₂×#13108075	C₄	semidirect
46. C₂×D₄₀₄	C₂×C₄	semidirect
63. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
64. C₂×C₄×D₁₀₁	C₂×C₄	semidirect
74. C₂×Q₈×C₁₀₁	C₂×C₄	semidirect
112. C₂×C₄×C₁₀₁	#41	semidirect
124. C₂×Dic₁₀₁	#41	semidirect
126. D₄₀₄	C₂²⋊C₄	semidirect
171. C₁₀₁	#2339	semidirect
176. #391	D₁₀₁	semidirect
216. C₂×Q₈	C₄×D₁₀₁	semidirect
301. C₂×C₄	#13108075	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13108075	2
C₂×C₄×D₄₀₄	2
C₂×#15692217	2
#20265084	2
#20265055	2
#20265039	2
#20265034	2
C₄×#13108051	2
#95×D₁₀₁	2
#391×C₁₀₁	2
#2339	101